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If 20 per cent managers are technocrats, the probability that a random committee of 5 managers consists of exactly 2 technocrats is
- a)0.2048
- b)0.4000
- c)0.4096
- d)0.9421
Correct answer is option 'A'. Can you explain this answer?
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To solve this problem, we need to apply the concept of probability and combinations.
Given:
- 20% of managers are technocrats, which means that out of 100 managers, 20 are technocrats.
- We need to find the probability that a random committee of 5 managers consists of exactly 2 technocrats.
Step 1: Calculate the number of ways to choose 2 technocrats
To calculate the number of ways to choose 2 technocrats out of 20, we can use the combination formula:
nCr = n! / ((n - r)! * r!)
In this case, n = 20 (total number of technocrats) and r = 2 (number of technocrats to be chosen).
So, the number of ways to choose 2 technocrats = 20C2 = 20! / ((20 - 2)! * 2!) = (20 * 19) / (2 * 1) = 190.
Step 2: Calculate the number of ways to choose 3 non-technocrats
Since we need a committee of 5 managers consisting of exactly 2 technocrats, the remaining 3 managers must be non-technocrats.
To calculate the number of ways to choose 3 non-technocrats out of 80 (100 - 20), we can use the same combination formula:
nCr = n! / ((n - r)! * r!)
In this case, n = 80 (total number of non-technocrats) and r = 3 (number of non-technocrats to be chosen).
So, the number of ways to choose 3 non-technocrats = 80C3 = 80! / ((80 - 3)! * 3!) = (80 * 79 * 78) / (3 * 2 * 1) = 82,160.
Step 3: Calculate the total number of ways to choose the committee of 5 managers
To calculate the total number of ways to choose a committee of 5 managers out of 100, we can again use the combination formula:
nCr = n! / ((n - r)! * r!)
In this case, n = 100 (total number of managers) and r = 5 (number of managers to be chosen).
So, the total number of ways to choose a committee of 5 managers = 100C5 = 100! / ((100 - 5)! * 5!) = (100 * 99 * 98 * 97 * 96) / (5 * 4 * 3 * 2 * 1) = 75,287,520.
Step 4: Calculate the probability
The probability of choosing exactly 2 technocrats in a committee of 5 managers can be calculated by dividing the number of ways to choose 2 technocrats and 3 non-technocrats by the total number of ways to choose 5 managers.
Probability = (Number of ways to choose 2 technocrats * Number of ways to choose 3 non-technocrats) / Total number of ways to choose 5 managers
Probability = (190 * 82,160) / 75,287,520
Probability ≈ 0.2048
Therefore, the correct answer is option A) 0.2048.
Given:
- 20% of managers are technocrats, which means that out of 100 managers, 20 are technocrats.
- We need to find the probability that a random committee of 5 managers consists of exactly 2 technocrats.
Step 1: Calculate the number of ways to choose 2 technocrats
To calculate the number of ways to choose 2 technocrats out of 20, we can use the combination formula:
nCr = n! / ((n - r)! * r!)
In this case, n = 20 (total number of technocrats) and r = 2 (number of technocrats to be chosen).
So, the number of ways to choose 2 technocrats = 20C2 = 20! / ((20 - 2)! * 2!) = (20 * 19) / (2 * 1) = 190.
Step 2: Calculate the number of ways to choose 3 non-technocrats
Since we need a committee of 5 managers consisting of exactly 2 technocrats, the remaining 3 managers must be non-technocrats.
To calculate the number of ways to choose 3 non-technocrats out of 80 (100 - 20), we can use the same combination formula:
nCr = n! / ((n - r)! * r!)
In this case, n = 80 (total number of non-technocrats) and r = 3 (number of non-technocrats to be chosen).
So, the number of ways to choose 3 non-technocrats = 80C3 = 80! / ((80 - 3)! * 3!) = (80 * 79 * 78) / (3 * 2 * 1) = 82,160.
Step 3: Calculate the total number of ways to choose the committee of 5 managers
To calculate the total number of ways to choose a committee of 5 managers out of 100, we can again use the combination formula:
nCr = n! / ((n - r)! * r!)
In this case, n = 100 (total number of managers) and r = 5 (number of managers to be chosen).
So, the total number of ways to choose a committee of 5 managers = 100C5 = 100! / ((100 - 5)! * 5!) = (100 * 99 * 98 * 97 * 96) / (5 * 4 * 3 * 2 * 1) = 75,287,520.
Step 4: Calculate the probability
The probability of choosing exactly 2 technocrats in a committee of 5 managers can be calculated by dividing the number of ways to choose 2 technocrats and 3 non-technocrats by the total number of ways to choose 5 managers.
Probability = (Number of ways to choose 2 technocrats * Number of ways to choose 3 non-technocrats) / Total number of ways to choose 5 managers
Probability = (190 * 82,160) / 75,287,520
Probability ≈ 0.2048
Therefore, the correct answer is option A) 0.2048.
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